Dynamic covariance estimation for multivariate time series suffers from the curse of dimensionality which renders parsimonious estimation methods essential for conducting reliable statistical inference. We address this problem by modeling the underlying volatility dynamics of a time series vector through a lower dimensional collection of latent time-varying stochastic factors. Furthermore, we apply a Normal-Gamma prior to the elements of the factor loadings matrix. This hierarchical shrinkage prior is a generalization of the Bayesian lasso and effectively shrinks the factor loadings of unimportant factors towards zero, thereby increasing parsimony even more. Estimation is carried out via Bayesian MCMC methods that allow to obtain draws from the high-dimensional posterior distribution. To guarantee efficiency of the samplers, we utilize several variants of an ancillarity-sufficiency interweaving strategy (ASIS) for sampling the factor loadings. We implement the sampler in a compiled programming language which is interfaced to R for increased usability. Through extensive simulation studies, we demonstrate the effectiveness of the shrinkage prior for sparse loadings matrices. Furthermore, we apply the model to 5000 daily log-returns of 300 stocks listed in the S&P 500 index. This is still work in progress.

In this talk we will discuss some relations between percolation on a given graph G and its geometry. There are several intresting questions relating various properties of G, such as growth or dimension, and the process of percolation on G. In particular one could look for conditions under which the critical percolation threshold p_c(G) is non-trivial, that is: p_c(G) is strictly between zero and one. In a very importante paper on this subject, Benjamini and Schramm asked whether it is true that for every graph satisfying dim(G) > 1, one has p_c(G) < 1. We will explain this question in detail, explaining what they meant by the dimension of a graph and we will present a result that has recently been obtained in this direction.

One of the greatest challenges in the fields of quantum information processing and quantum technologies is the detailed coherent control over each and all of the constituents of quantum systems with an ever increasing number of particles. Within this endeavor, the harnessing of many-body entanglement against the detrimental effects of the environment is a major and pressing issue. Besides being an important concept from a fundamental standpoint, entanglement has been recognized as a crucial resource for quantum speed-ups or performance enhancements over classical methods. Understanding and controlling many-body entanglement in open systems may have strong implications in quantum computing, quantum simulations of many-body systems, quantum cryptography, quantum metrology, our understanding of the quantum-to-classical transition, and other important questions of quantum foundations.
In this seminar entanglement will be taken as a dynamic quantity on its own, that evolves due to the unavoidable interaction of the entangled system with its surroundings. I will introduce the main aspects of entanglement dynamics in open quantum systems, portraying its richness and complexity. After setting the stage, I will present two different approaches two deal with entanglement dynamics: First, for bipartite systems I'll present a deterministic dynamical equation for entanglement. Second, in order to cope with many-body systems, I'll resort to a statistical description of typical entanglement dynamics. The latter relies solely on geometrical aspects of the space of states.

In this talk we will introduce a continuous time Markov process which is known as the symmetric simple exclusion process (SSEP) with a disorder at a bond. The process will be evolving on the one dimensional discrete torus Tn with n sites. We attach a clock to each bond of Tn, all the clocks being independent and exponential distributed with parameter 1. After a ring of a clock, the particles at the bonds exchange positions. We perturb this dynamics by introducing a bond disorder. For that purpose, we only change the parameter of the clock corresponding to the jumps between the sites −1 and 0, and we take it equal to α/nβ, where α > 0 and β ≥ 0. As a consequence, microscopically, as beta increases the more difficult is the passage of particles across the bond [−1,0]. We will present the hydrodynamics for this model which are given by the heat equation with periodic, Robin’s or Neumann’s boundary conditions depending on the range of β. We will also present several phase transitions which appear by changing the value of α.